Existence and nonexistence of normalized solutions for nonlinear Schr\"{o}dinger equation involving combined nonlinearities in bounded domain

TL;DR

This paper investigates the existence, multiplicity, and nonexistence of normalized solutions for a nonlinear Schrödinger equation with combined nonlinearities in bounded domains, revealing conditions under which solutions exist or do not.

Contribution

It provides new results on the existence and nonexistence of solutions for the nonlinear Schrödinger equation with combined nonlinearities, including a dichotomy for the Brézis-Nirenberg problem.

Findings

Existence of local minimizers for small prescribed norms.

Existence of mountain pass solutions under certain conditions.

Nonexistence of solutions for large prescribed norms in specific domains.

Abstract

In this paper, we consider the existence, multiplicity and nonexistence of solutions for the following equation \begin{equation*} \begin{cases} \begin{aligned} &-\Delta u+\omega u=\mu u^{p-1}+u^{q-1},~ u>0 \quad &&\text { in } \Omega, \\ &u=0 &&\text { on } \partial\Omega, \\ \end{aligned} \end{cases} \end{equation*} with prescribed $L^2$-norm $\|u\|_2^2=\rho$, where $N\ge 1$, $\rho>0$, $\mu\in \mathbb{R}$, $1<p\le q$, and $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain. The parameter $\omega\in\mathbb{R}$ arises as a Lagrange multiplier. Firstly, when $2<p\le q\le \frac{2N}{(N-2)^+}$ and $\rho$ is small, we establish the existence of a local minimizer of energy. Furthermore, when $\mu\ge 0$ and $\Omega$ is a star-shaped domain, using the monotonicity trick and the Pohozaev identity, we show that there exists a second solution which is of mountain pass type. Secondly, when $\mu\ge 0$, $N\ge 3$, $1<p\le 2$, $q\ge \max\left\{\frac{2N}{N-2}, 3\right\}$ and $\Omega$ is a convex domain, using the moving-plane method, we prove the nonexistence of normalized solutions for large $\rho$. Finally, when $\mu=0$, $N\ge 3$, $q=\frac{2N}{N-2}$ and $\Omega$ is a ball, we give a dichotomy result of normalized solutions for the Br\'{e}zis-Nirenberg problem by continuation arguments.